Clavius's Law/Formulation 1

Theorem

$\neg p \implies p \vdash p$

Proof 1

By the tableau method of natural deduction:

$\neg p \implies p \vdash p$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg p \implies p$ Premise (None)
2 $p \lor \neg p$ Law of Excluded Middle (None)
3 3 $\neg p$ Assumption (None) Either $p$ is false ...
4 1, 3 $p$ Modus Ponendo Ponens: $\implies \mathcal E$ 1, 3
5 5 $p$ Assumption (None) ... or $p$ is true
6 1 $p$ Proof by Cases: $\text{PBC}$ 2, 3 – 4, 5 – 5 Assumptions 3 and 5 have been discharged

$\blacksquare$

Proof 2

By the tableau method of natural deduction:

$\neg p \implies p \vdash p$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg p \implies p$ Premise (None)
2 2 $p \implies \bot$ Assumption (None)
3 2 $\neg p$ Sequent Introduction 2 Negation as Implication of Bottom
4 1,2 $p$ Modus Ponendo Ponens: $\implies \mathcal E$ 1, 3
5 1 $(p \implies \bot) \implies p$ Rule of Implication: $\implies \mathcal I$ 2 – 4 Assumption 2 has been discharged
6 1 $p$ Sequent Introduction 5 Peirce's Law

$\blacksquare$

Source of Name

This entry was named for Christopher Clavius.